In mathematics, specifically in category theory, Day convolution is an operation on functors that can be seen as a categorified version of function convolution. It was first introduced by Brian Day in 1970 in the general context of enriched functor categories.

Day convolution gives a symmetric monoidal structure on H o m ( C , D ) {\displaystyle \mathrm {Hom} (\mathbf {C} ,\mathbf {D} )} for two symmetric monoidal categories C , D {\displaystyle \mathbf {C} ,\mathbf {D} }.

Another related version is that Day convolution acts as a tensor product for a monoidal category structure on the category of functors [ C , V ] {\displaystyle [\mathbf {C} ,V]} over some monoidal category V {\displaystyle V}.

Definition

First version

Given F , G : C → D {\displaystyle F,G\colon \mathbf {C} \to \mathbf {D} } for two symmetric monoidal C , D {\displaystyle \mathbf {C} ,\mathbf {D} }, we define their Day convolution as follows.

It is the left kan extension along C × C → ⊗ C {\displaystyle \mathbf {C} \times \mathbf {C} \to ^{\otimes }\mathbf {C} } of the composition C × C → F , G D × D → ⊗ D {\displaystyle \mathbf {C} \times \mathbf {C} \to ^{F,G}\mathbf {D} \times \mathbf {D} \to ^{\otimes }\mathbf {D} }

Thus evaluated on an object O ∈ C {\displaystyle O\in \mathbf {C} }, intuitively we get a colimit in D {\displaystyle \mathbf {D} } of F ( x ) ⊗ G ( y ) {\displaystyle F(x)\otimes G(y)} along approximations of O ∈ C {\displaystyle O\in \mathbf {C} } as a pure tensor x ⊗ y {\displaystyle x\otimes y}

Left kan extensions are computed via coends, which leads to the version below.

Enriched version

Let ( C , ⊗ c ) {\displaystyle (\mathbf {C} ,\otimes _{c})} be a monoidal category enriched over a symmetric monoidal closed category ( V , ⊗ ) {\displaystyle (V,\otimes )}. Given two functors F , G : C → V {\displaystyle F,G\colon \mathbf {C} \to V}, we define their Day convolution as the following coend.

F ⊗ d G = ∫ x , y ∈ C C ( x ⊗ c y , − ) ⊗ F x ⊗ G y {\displaystyle F\otimes _{d}G=\int ^{x,y\in \mathbf {C} }\mathbf {C} (x\otimes _{c}y,-)\otimes Fx\otimes Gy}

If ⊗ c {\displaystyle \otimes _{c}} is symmetric, then ⊗ d {\displaystyle \otimes _{d}} is also symmetric. We can show this defines an associative monoidal product:

( F ⊗ d G ) ⊗ d H ≅ ∫ c 1 , c 2 ( F ⊗ d G ) c 1 ⊗ H c 2 ⊗ C ( c 1 ⊗ c c 2 , − ) ≅ ∫ c 1 , c 2 ( ∫ c 3 , c 4 F c 3 ⊗ G c 4 ⊗ C ( c 3 ⊗ c c 4 , c 1 ) ) ⊗ H c 2 ⊗ C ( c 1 ⊗ c c 2 , − ) ≅ ∫ c 1 , c 2 , c 3 , c 4 F c 3 ⊗ G c 4 ⊗ H c 2 ⊗ C ( c 3 ⊗ c c 4 , c 1 ) ⊗ C ( c 1 ⊗ c c 2 , − ) ≅ ∫ c 1 , c 2 , c 3 , c 4 F c 3 ⊗ G c 4 ⊗ H c 2 ⊗ C ( c 3 ⊗ c c 4 ⊗ c c 2 , − ) ≅ ∫ c 1 , c 2 , c 3 , c 4 F c 3 ⊗ G c 4 ⊗ H c 2 ⊗ C ( c 2 ⊗ c c 4 , c 1 ) ⊗ C ( c 3 ⊗ c c 1 , − ) ≅ ∫ c 1 , c 3 F c 3 ⊗ ( G ⊗ d H ) c 1 ⊗ C ( c 3 ⊗ c c 1 , − ) ≅ F ⊗ d ( G ⊗ d H ) {\displaystyle {\begin{aligned}&(F\otimes _{d}G)\otimes _{d}H\\[5pt]\cong {}&\int ^{c_{1},c_{2}}(F\otimes _{d}G)c_{1}\otimes Hc_{2}\otimes \mathbf {C} (c_{1}\otimes _{c}c_{2},-)\\[5pt]\cong {}&\int ^{c_{1},c_{2}}\left(\int ^{c_{3},c_{4}}Fc_{3}\otimes Gc_{4}\otimes \mathbf {C} (c_{3}\otimes _{c}c_{4},c_{1})\right)\otimes Hc_{2}\otimes \mathbf {C} (c_{1}\otimes _{c}c_{2},-)\\[5pt]\cong {}&\int ^{c_{1},c_{2},c_{3},c_{4}}Fc_{3}\otimes Gc_{4}\otimes Hc_{2}\otimes \mathbf {C} (c_{3}\otimes _{c}c_{4},c_{1})\otimes \mathbf {C} (c_{1}\otimes _{c}c_{2},-)\\[5pt]\cong {}&\int ^{c_{1},c_{2},c_{3},c_{4}}Fc_{3}\otimes Gc_{4}\otimes Hc_{2}\otimes \mathbf {C} (c_{3}\otimes _{c}c_{4}\otimes _{c}c_{2},-)\\[5pt]\cong {}&\int ^{c_{1},c_{2},c_{3},c_{4}}Fc_{3}\otimes Gc_{4}\otimes Hc_{2}\otimes \mathbf {C} (c_{2}\otimes _{c}c_{4},c_{1})\otimes \mathbf {C} (c_{3}\otimes _{c}c_{1},-)\\[5pt]\cong {}&\int ^{c_{1},c_{3}}Fc_{3}\otimes (G\otimes _{d}H)c_{1}\otimes \mathbf {C} (c_{3}\otimes _{c}c_{1},-)\\[5pt]\cong {}&F\otimes _{d}(G\otimes _{d}H)\end{aligned}}}

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